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For the H-atom, how does the Atomic model as described by Bohr connects to the atomic model as described by Heisenberg/Schroedinger?

For the H-atom, how does the Atomic model as described by Bohr connects to the atomic model as described by Heisenberg/Schroedinger?

Physical Chemistry
Physical Chemistry
2nd Edition
ISBN:9781133958437
Author:Ball, David W. (david Warren), BAER, Tomas
Publisher:Ball, David W. (david Warren), BAER, Tomas
Chapter11: Quantum Mechanics: Model Systems And The Hydrogen Atom
Section: Chapter Questions
Problem 11.14E
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For the H-atom, how does the Atomic model as described by Bohr connects to the atomic model as described by Heisenberg/Schroedinger?
206 Chapter 6 The Hydrogen Atom expression for Î, in spherical coordinates is so much simpler than for θ or Î, (cf. Equation 6.37). Clearly the rotating system does not know x from y from z and, in fact, this inability to distinguish between the three directions explains the (21 + 1)-fold degeneracy. 6-4. Hydrogen Atomic Orbitals Depend upon Three Quantum Numbers Up to now we have solved Equation 6.9, giving the angular part of the hydrogen atomic orbitals. Now we will solve Equation 6.8, giving the radial part of the hydrogen atomic orbitals. Equation 6.8 with ß set equal to 1 (1+1) can be written as dR [ħ²1(1+1) e² -2m² dr (dr) + [²+¹) - [25-B]R(r) = ER(r)=0 2m¸r² E Equation 6.43 is an ordinary differential equation in r. It is somewhat tedious to solve, but once solved, we find that for solutions to be acceptable as the wave functions, the energy must be quantized according to me4 8h²n² R₁(r) = - meª 32л²εħ²n² E = If we introduce the Bohr radius from Section 1-8, a = = εh²/лm₂ е² = 4πεħ²/me², then Equation 6.44 becomes 1/2 31 (n-1-1)! 2n[(n+1)!]³ e² 8πε an? It is surely remarkable that these are the same energies obtained from the Bohr model of the hydrogen atom. Of course, the electron now is not restricted to the sharply defined orbits of Bohr but is described by its wave function, (r, 0, 0). In the course of solving Equation 6.43, we find not only that an integer n occurs naturally but that n must satisfy the condition that n ≥ 1 + 1, which is usually written as 0<l<n-1 n = 1, 2, ... (6.46) because we have already seen that the smallest possible value of 1 is zero. (Equation 6.46 might be familiar from general chemistry.) The solutions to Equation 6.43 depend on two quantum numbers n and I and are given by 2 nao. n = 1, 2, ... 230 n = 1, 2, ... 1+3/2 re (6.43) 2r nao (6.44) (6.45) (6.47) where the L2+(2r/na) are called associated Laguerre polynomials. The first few associated Laguerre polynomials are given in Table 6.4.
Transcribed Image Text:206 Chapter 6 The Hydrogen Atom expression for Î, in spherical coordinates is so much simpler than for θ or Î, (cf. Equation 6.37). Clearly the rotating system does not know x from y from z and, in fact, this inability to distinguish between the three directions explains the (21 + 1)-fold degeneracy. 6-4. Hydrogen Atomic Orbitals Depend upon Three Quantum Numbers Up to now we have solved Equation 6.9, giving the angular part of the hydrogen atomic orbitals. Now we will solve Equation 6.8, giving the radial part of the hydrogen atomic orbitals. Equation 6.8 with ß set equal to 1 (1+1) can be written as dR [ħ²1(1+1) e² -2m² dr (dr) + [²+¹) - [25-B]R(r) = ER(r)=0 2m¸r² E Equation 6.43 is an ordinary differential equation in r. It is somewhat tedious to solve, but once solved, we find that for solutions to be acceptable as the wave functions, the energy must be quantized according to me4 8h²n² R₁(r) = - meª 32л²εħ²n² E = If we introduce the Bohr radius from Section 1-8, a = = εh²/лm₂ е² = 4πεħ²/me², then Equation 6.44 becomes 1/2 31 (n-1-1)! 2n[(n+1)!]³ e² 8πε an? It is surely remarkable that these are the same energies obtained from the Bohr model of the hydrogen atom. Of course, the electron now is not restricted to the sharply defined orbits of Bohr but is described by its wave function, (r, 0, 0). In the course of solving Equation 6.43, we find not only that an integer n occurs naturally but that n must satisfy the condition that n ≥ 1 + 1, which is usually written as 0<l<n-1 n = 1, 2, ... (6.46) because we have already seen that the smallest possible value of 1 is zero. (Equation 6.46 might be familiar from general chemistry.) The solutions to Equation 6.43 depend on two quantum numbers n and I and are given by 2 nao. n = 1, 2, ... 230 n = 1, 2, ... 1+3/2 re (6.43) 2r nao (6.44) (6.45) (6.47) where the L2+(2r/na) are called associated Laguerre polynomials. The first few associated Laguerre polynomials are given in Table 6.4.
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